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2012
REVIEW ON SOLVING THE JOB SHOP SCHEDULING PROBLEM: RECENT
DEVELOPMENT AND TRENDS
Ivan Lazár, M.Sc.,
Faculty of Manufacturing Technologies
Technical University of Kosice with a seat in
Presov,
Bayerova 1, 080 01 Prešov,
Slovak Republic
[email protected]
Abstract
This article discusses the development of
the Job Shop problem
and
the
development of methods
to
be used in
solving JSSP. It
also defines the
groups JSS
Problems, which
are
divided according
to
the complexity of the solution. The article includes
the evaluation of publications research and methods
research that are used in various publications.
Key words: job shop, scheduling problems, process
layout
INTRODUCTION
The job-shop problem is to schedule a set
of jobs on a set of machines, subject to the
constraint that each machine can handle at most one
job at a time and the fact that each job has a
specified processing order through the machines.
The objective is to schedule the jobs so as to
minimize the maximum of their completion times.
This problem is not only NP-hard it is also
has the well-earned reputation of being one of the
most computationally stubborn combinatorial
problems considered to date. This intractability is
one of the reasons the problem has been so widely
studied. Indeed, some of the excitement in working
on the problem no doubt arose from the fact that a
specific instance, with 10 machines and 10 jobs,
dealt in a book by Muth and Thompson [16]
remained unsolved for over 20 years. This
particular instance was finally settled in 1985 by
Carlier and Pinson [2], [3] that incorporates the
ideas of Lageweg et. al [9].
The remainder of the paper is structured as
follows: The next section outlines research
methodology employed. Then, a position of the Job
Shop production properties in the context of layout
design is analyzed. The fourth section focuses on
defining of Job Shop Scheduling Problem (JSSP).
In the further section, definitions and a
classification
of
job-shops
is
presented.
Subsequently, a review of frequent approaches and
methods for JSSP solving is treated. A final section
of the paper summarizes findings of this review on
mapping the field over the last 5 years.
RESEARCH METHODOLOGY
Usually quantitative research is based on a
large representative survey and its outcomes are
reliable data that can be generalized. As it has been
mentioned above, the survey of the Job shop
scheduling problem is aimed to map the field over
the last 5 years and is focused:

to quantify of research effort in developing
a wide range of approaches and methods
for solving JSSP
 and to determine which of these methods
are most commonly used in solving this
problem.
In this quantitative research were
used especially scientific articles registered in
Science Direct and Scirus. Keywords applied in
search engines were: job shop, scheduling
problems, heuristics and metaheuristics methods,
genetic algorithm, tabu search, local search, linear
and dynamic programming, etc.
Relevant findings compiled from the
literature review are graphically shown on Figures
2 and 3 and also briefly commented in the final
sections.
POSITION
OF
THE
PRODUCTION SYSTEM
JOB
SHOP
According to Groover [7] there are three
types of production associated with discreteproduct manufacture:
1. Job shop production (low volume production),
2. Batch production (Medium- sized lots of the
same item or product),
3. Mass production (two categories of mass
production can be distinguished):
a) quantity production (production of simple
single parts such as screws),
b) flow production (production of complex single
parts such as automotive engine blocks) 7.
This classification can also serve for plants
used in the process industries:
For each type of production is more or less
suitable one of the four principal types of plant
layout:
I) Fixed-position layout (large unites, such as a
ship)
II) Process layout (according to you, Technologyoriented manufacturing)
III) Product-flow layout (according to you Objectoriented manufacturing)
Typical layouts for the given types of
production are specified in Table 1.
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Table 1 Possible layouts for the given types of
production
Type of plant layout
Fixed-position layout
Process layout
Product-flow layout
Type of production
Job shop production
Job shop production
Batch production
Batch production
Mass production
Colored boxes demarcate a field of the
cellular
manufacturing
systems.
Thus
manufacturing system designers have a dilemma for
batch production in deciding when to apply Process
layout and when to apply for the same one Productflow layout. This issue can be solved through
testing algorithm presented by Modrak [14].
Cellular manufacturing is currently a topical issue
especially for manufacturers due to fact that process
layout in case of batch production is not enough to
pass customer requirements [15]. Accordingly, for
many manufacturers, the actual issue is
transformation of Process layouts to Cellular
layouts as it is shown in Figure 1.
Then Ck = Sk + pk is the finishing time of k
and (Sk ) defines a schedule. A schedule (Sk ) is
feasible if for any succeeding operations k and s (k)
of the same job Sk + pk ≤ Ss(k) holds and for two
operations k and h with M (k) = M (h)either Sk+pk ≤
Sh or Sh + ph ≤ Sk. One has to find a feasible
schedule (Sk) which minimizes the makespan
.
Lageweg et al. in 1981 developed a computer
program
MSPCLASS
for
an
automatic
classification of scheduling problems. This program
based on the α-classification scheme calculates
problems which are:
 maximal polynomially solvable: the
hardest problems which are polynomially
solvable, i.e. problems which are known to
be polynomially solvable, but any harder
cases are not known to be polynomially
solvable,




Figure 1 Transformation from Process layout (a) to
Cellular layout (b)
DEFINITION AND CLASSIFICATION OF
THE JOB-SHOP PROBLEM
The job-shop problem can be formulated
as follows. Given are m machines M1,M2, ..., Mm
and n jobs J1,J2,..., Jn. Job Jj consists of nj
operations Oij (i = 1, ..., nj )which have to be
processed in the order O1j O2j , ..., Onj j. It is
convenient to enumerate all operations of all jobs
by k = 1, ..., N where
. For each
operation k = 1, ..., N we have a processing time
pk> 0 and a dedicated machine M(k). k must be
processed for pk time units without preemptions on
M(k). Additionally a dummy starting operation 0
and a dummy finishing operation N + 1, each with
zero processing time, are introduced. We assume
that for two succeeding operations k = Oij and s (k)
= Oi+1,j of the same job M (k)≠ M (s (k)) holds. Let
Sk be the starting time of operation k.
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maximal pseudopolynomially solvable: the
hardest problems which are known to be
pseudopolynomially
(but
not
polynomially) solvable,
minimal NP-hard: the easiest problems
which are NP-hard, i.e. problems which
are known to be NP-hard, but any easier
cases are not known to be NP-hard,
minimal open: problems for which the
complexity status is not known, but all
easier cases are known to be polynomially
solvable,
maximal open: problems for which the
complexity status is not known, but all
harder cases are known to be NP-hard [9].
Table 2 Job-shop problems with preemption
Job-shop problems with preemption
Maximal polynomially solvable
Sotskov
(1991)
Sotskov
(1991)
Maximal pseudopolynomially solvable
Middendorf &
Timkovsky
(1999)
Middendorf &
Timkovsky
(1999)
Minimal NP-hard
Brucker et
al. (1999B)
Lenstra &
Rinnooy
Kan (1979)
Brucker et
al. (1999B)
Lenstra (-)
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Table 3 Job-shop problems without preemption
Job-shop problems without preemption
Maximal polynomially solvable
Timkovsky
(1997)
Kubiak&Timko
vsky (1996)
Kravchenko
(1999A)
Brucker&
Kraemer (1996)
Sotskov (1991)
Brucker et al.
(1997A)
Brucker&
Kraemer (1996)
Sotskov (1991)
Brucker et al.
(1997A)
Maximal pseudopolynomially solvable
Middendorf&Ti
mkovsky (1999)
Kravchenko
(1999A)
Middendorf&Ti
mkovsky (1999)
Minimal NP-hard
Sotskov&Shakhlevi
ch (1995)
Lenstra&RinnooyK
an (1979)
Timkovsky (1985)
Lenstra&RinnooyK
an (1979)
Timkovsky (1998)
Sotskov&Shakhlevi
ch (1995)
Garey et al. (1976)
Timkovsky (1998)
Lenstra (-)
Timkovsky (1998)
Timkovsky (1998)
Timkovsky (1998)
Kravchenko (1999)
Timkovsky (1998)
Table 4 Job-shop problems with no-wait
Job-shop problems with no-wait
Maximal polynomially solvable
Kravchenko
(1998)
Baptiste et al.
(2004)
Baptiste et al.
(2004)
Maximal pseudopolynomially solvable
Timkovsky
Kubiak (1989)
Minimal NP-hard
Timkovsky
Kubiak (1989)
Timkovsky
(1998)
Sahni& Cho
(1979)
Sriskandarajah&L
adet (1986)
Timkovsky
(1998)
Timkovsky
(1998)
Roeck (1984)
Timkovsky
(1998)
Sriskandarajah&L
adet (1986)
Timkovsky
(1998)
Timkovsky
(1998)
APPROACHES AND METHODS TO SOLVE
JSSP
Brucker and Schile [21] were the first
authors to describe this problem in 1990. They
developed a polynomial graphical algorithm for a
two-job problem. In the scheduling of job-shops,
the most common methodology is materials
requirement planning (MRP) [1]. However, MRP is
mostly a planning tool and is not really designed for
detailed-level scheduling. In many companies,
scheduling is performed by experienced shop-floor
personnel with pencil, paper, a few graphical aids
(such as Gantt chart) and perhaps a modern
industrial database [6], [10]. Simple dispatching
rules are often used for solving immediate
problems, such as sequencing at the work-center
level. The result can be scheduling chaos, where
completion dates cannot be predicted and work-inprocess (WIP) inventory builds [10]. Sometimes,
even high-level management must chase down
high-priority jobs on the shop floor. Many
dispatching rules have been presented and
implemented based on due dates, criticality of
operations, processing times, and resource
utilization. The „critical ratio“ defined by one
definition as a ratio of remaining processing time
over remaining time to due date, has been very
popular in job-shops [6]. More complicated
heuristics take into account some combination of
the above factors. For example, Viviers´ algorithm
incorporates three priority classes in the shortest
processing tim (SPT) rule [20]. Each job is assigned
an index equal to its processing time plus a value
graded to its priority class. High-priority jobs have
low index values and are processed first according
to the SPT rule. Heuristics have been comparatively
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evaluated. Many artificial intelligence (AI)
approaches also use dispatching rules or heuristics
for scheduling [5], [8]. It is generally very difficult
to evaluate the performance of schedules generated
by these methods. The result may also depend upon
the initial ordering of jobs. This implies that minnor
changes in jobs and/or resource availability from
one day to the next may result in quite different
schedules. There has been a great deal of effort
concentrated on optimalization methodologies.
However, exact algorithms are not
effective for solving JSSP and large instances.
Several heuristic procedures have been developed
in recent years for the JSSP [17]. The methods in
this category include dynamic programming and the
branch-and-bound method, simulated annealing
(SA) and genetic algorithm (GA) [4]. Because the
number of possible sequences growts exponentially
as the problem size, these methods become very
computational intensive for even small-sized jobshops [9]. There are a plenty heuristic procedures
and rules to assist in this endeavour. However, rules
leading to the optimum schedule have been elusive.
The predominant scheduling methods now used are
specifically tailored to the type of job shop.
Generally, various rules are tried and those
giving the best result are used as a starting point.
The human expert sifts the schedule through his
experience filter, negotiates with affected parties,
and finalizes a schedule. Expert systems are
beginning to impact in this area. By assuming some
of the filtering and negotiating roles of the human
expert, they can allow schedulers to look at more
alternatives and/or produce more timely schedules.
However, the meta-heuristics methods
have led to better results than the traditional
dispatching or greedy heuristic algorithm. JSSP
could be turned into the Job-shop scheduling
problem when a routing is chosen, so when solving
JSSP, hierarchical approach and integrated
approach have been used [11], [12]. The
hierarchical approach could reduce difficulty by
decomposing the JSSP into a sequence of subproblems. Dauzère-Pérès and Paulli [19] solved the
routing subproblem using some existing
dispatching rules, and then solved the scheduling
sub-problem by different tabu search methods.
Integrated approach could achieve better results,
but it is rather difficult to be implemented in real
operations. Mati, et al [13] proposed different tabu
search heuristic approach to solve the JSSP using
an
integrated
approach.
Mastrolilli
and
Gambardella [11] proposed some neighborhood
functions for the JSSP, which can be used in metaheuristic optimization techniques, and achieve
better computational results than any other heuristic
developed so far, both in terms of computational
time and solution quality. GA is an effective metaheuristic to solve combinatorial optimization
58
2012
problems, and has been successfully adopted to
solve the JSSP.
Table 5 Survey of the JSSP problem approaches
AUTHORS
F. Pezzella, G.
Morganti, G.
Ciaschett
Jin-hui Yang, et. al.
YEAR
MET
SIZE
2008
GA
20x15
2008
MA
10x10
Márton Drótos, et. al.
Guan-Chun Luh,
Chung-Huei Chueh
Gromicho, et.al.
Shi Qiang Liu, Erhan
Kozan
Christian Artigues, et.
al
Ce´sar Rego, Renato
Duarte
Shijin Wang, Jianbo
Yu
2009
N-A
N-A
2009
IA
30x10
2009
DP
N-A
2009
SBP
N-A
2009
B and
B
N-A
2009
F&F
15x15
2010
H
10x10
2010
CDR,
NRGA
30x10
2010
H
30x10
2010
GA
30x10
2010
GA
20x20
2010
SAA
50x10
E. Moradi, et. al.
Shijin Wang, Jianbo
Yu
Leila Asadzadeh,
Kamran Zamanifa
L. De Giovanni, F.
Pezzella
Rui Zhang, Cheng
Wu
Deming Lei
Wannaporn Teekeng,
Arit Thammano
Guohui Zhang, Liang
Gao,Yang Shi
2010
2011
2011
RKG
A
FrL,
FL
GA
15x10
20x15
20x15
Rubiyah Yusof, et. al.
2011
Veronique Sels, et. al.
Marnix Kammer, et.
al.
Yazid Mati, et. al.
R.TavakkoliMoghaddam, et. al.
Darrell F. Lochtefeld,
Frank W. Ciarallo
2011
MICR
O GA
MH
2011
LS
20x20
2011
LS
50x8
2011
PSO
100x20
Min Liu, et. al.
2011
Liang Gao, et. al.
2011
MOE
As
GA,
SA
MA
Mati, Yazid, et. al.
2011
LS
10x10
Ye Li , Yan Che
Rui Zhang, Cheng
Wu
Moslehi Ghasem,
Mahnam Mehdi
2011
TPA
20x10
2011
TSA
N-A
Jianchao Tang, et. al.
2011
-
2011
2011
PSO,
LS
PSO,
GA
30x10
2x20
50x10
10x10
15x15
N-A
20x15
MET-Methods
(MOEAs)-Multiple Objective Evolutionary
Algorithms,
(CDR)-Composite Dispatching Rule,
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(NSGA-II)-Non Dominated Sort Genetic
Algorithm,
(NRGA)-Non Ranking Genetic Algorithm,
(SAA)-Simulated Annealing Algorithm,
(MA)-Memetic Algorithm,
(F&F)-Filter-and-Fan,
(TPA)-Team Process Algorithm,
(TSA)-Tree Search Algorithm,
(FrL)-Frog Leaping,(FL)-Fuzzy Logic,
(RKGA)-Random Key Genetic Algorithm,
(SA)-Simulated Annealing,
(SBP)-Shifting
Bottleneck
Procedure
algorithm,
(IA)-Immune Algorithm,
(DP)-Dynamic Programming
In the next graph in figure 2 are shown, as
consistent with the second objective of this study,
methods that are most commonly used in solving
this problem in order of frequency. The topicality
of genetic algorithm seems to be beyond a
reasonable doubt due to computational results
demonstrating the superiority in terms efficiency
and effectiveness. As regards to memetic
algorithms (MA), they represent one of the growing
areas of research in evolutionary computation and
are widely used as a synergy of evolutionary or any
population-based approach with separate individual
learning or local improvement procedures for
problem search.
One of the several approaches that may be
useful for the JSSP with the objective to minimize
the maximum completion time is particle swarm
optimization (PSO)-based memetic algorithm
(MA). In the PSO-based MA algorithm both PSObased searching operators and some special local
searching operators are employed to balance the
exploration and exploitation abilities. In particular,
this algorithm applies the evolutionary searching
mechanism of PSO, which is characterized by
individual improvement, population cooperation,
and competition to effectively perform exploration.
The second important group of algorithms
includes well-known Local search. Iterative LS is
powerful optimization procedures which have been
successfully applied to a number of JSSPs.
A general description of the pertinent findings
obtained from the presented survey is possible to
demonstrate by the following graphs.
In accordance with the methodology
section and the first objective of this study, one can
say that the number of articles dealing with the
JSSP began rapidly to expand starting from 2006
(see Figure 2 and 3). This can be perceived as
evidence that JSSP is quite popular research topic,
which is dealt by increasing number of authors.
2012
Figure 2 The most commonly used methods of
JSSP in order of frequency
Figure 3 Frequency of research articles dealing with
the JSSP over the last 5 years
SUMMARY
Based on the results of the presented study
can be stated that Job shop scheduling problem is
one of the topical problems in operations research,
which is continuously being updated in accordance
with the results of newest approaches.
The intention of this paper was to provide
an overview of one class of a large group of job
shop scheduling problem. In each its part
(especially in the previous section), is offered a
brief literature of works that dealt with the
particular approaches.
A part of the main
objectives of this study, considerable attention has
been paid to the concept of classification of job
shop problems and algorithms classification that are
pertinent to solve specified problems.
The results from the mapping of important
trends in developing new methods can be used as a
reference for future research in this area.
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